1. Introduction

With its fascinating properties and wide range of potential applications like negative index refraction [1], perfect absorber [2] and invisibility cloak [3–5 ], metamaterial (MM) has attracted considerable attention. Another key factor promoting its popularity is the well-developed transformation optics (TO), a good example of successful application of the fundamental principle that physical equation form is invariant under the coordinate transformation. MMs, coupled with TO, have paved a powerful, novel way to manipulate a physical field in a desired way. With the encouraging progress made in electromagnetic wave [3–9 ], MM and TO are quickly finding their way into other waves such as matter wave [10], acoustic wave [11] and elastic wave [12]. Recently MM and TO are successfully employed to manipulate other physical fields such as static magnetic field [13,14 ], current field [15,16 ], thermal field [17–22 ], diffusive mass [23–25 ] and electrostatic field [26].

Although great achievement has been made, MM and TO are usually limited to a single physical domain. Last year, bifunctional metamaterials and transformation multiphysics were proposed to achieve independent and simultaneous manipulation of multi-physics field [27]. The first successful experiment in multi-physics field was reported in cloaking thermal and electric fields simultaneously using artificial composite [28]. Our group proposed and demonstrated simultaneous and independent manipulation of electric and thermal fields using a bilayer structure [29]. We also have proposed a general method to obtain simultaneous manipulation of multi-physics field using both passive and active schemes [30]. Up to now, however, no work has been reported on the bifunctional concentrator that can concentrate multi-physics field simultaneously.

In this study, we propose a bifunctional concentrator that can concentrate electric and thermal fields simultaneously while keeping the external fields undistorted. Then we designed and tested a prototype. Numerical simulation and experimental results show good agreement, indicating the soundness and feasibility of our scheme.

2. Bifunctional concentrator

Suppose that the space is divided into three parts: core region ($r<a$), shell region ($a<r<b$) and exterior region ($r>b$) (see Fig. 1(a) ). The core region and exterior region are made of background material with thermal conductivity κ₀ and electric conductivity ${\sigma }_0$. According to S. Narayana’s work [17], to make a thermal concentrator (see Fig. 1(b)), the thermal conductivity of shell should be anisotropic and satisfy this relationship: ${\kappa }_{\theta }{\kappa }_r={\kappa }_0{}^2$, where ${\kappa }_r>{\kappa }_{\theta }$ (${\kappa }_r$ and ${\kappa }_{\theta }$ are thermal conductivity components of the shell in radial and circular directions, respectively). Similarly, for electric concentrator (see Fig. 1(c)), the electric conductivity of shell should satisfy this relationship: ${\sigma }_{\theta }{\sigma }_r={\sigma }_0{}^2$, where ${\sigma }_r>{\sigma }_{\theta }$ (${\sigma }_r$ and ${\sigma }_{\theta }$ are electric conductivity components of the shell in radial and circular directions, respectively). It is quite clear that to implement a shell satisfying these two conditions at the same time poses a great challenge. To accomplish this goal, one way is to employ bifunctional metamaterial [27, 28 ], which however suffers from great difficulties in fabrication. To overcome this problem, we utilized so-called fan-shaped structure shown in Fig. 2 , which is composed of alternating 18 wedges made of material A (ABS: ${\kappa }_{\text{A}}=0.15$ W/mK, ${\sigma }_{\text{A}}=0$ S/m) and 18 wedges made of material B (${\kappa }_{\text{B}}$ and ${\sigma }_{\text{B}}$). According to effective medium theory [31], the thermal and electric conductivity components can be expressed as:

 

Fig. 1 The principle for bifunctional concentrator: a) The corresponding physical model. The space is divided into three parts: I ($r<a$), II ($a<r<b$) and III ($r>b$). The thermal conductivity and electric conductivity of background material are κ₀ and ${\sigma }_0$, while the ones for the concentrator shell are κ ₁ and ${\sigma }_{\text{1}}$, respectively. b) The normalized thermal flux density distribution. The black lines represent thermal flux density vectors. c) The normalized electric current density distribution. The white lines represent current density vectors.

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Fig. 2 (a) The schematic illustration for practical realization of bifunctional concentrator. The concentrator shell is composed of alternating 18 wedges made of material A (ABS) and 18 wedges made of material B (aluminum). (b) The geometrical parameters: $a=6$ mm, $b=30$ mm. (c)The photograph of fabricated sample.

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Here, $f_A$and $f_B$ are the filling fraction for material A and material B ($f_A=f_B=0.5$). Then the above equations can be deduced as:

Assume that the background material is made of stainless steel with ${\kappa }_0=15$ W/mK and ${\sigma }_0=1.4\text{e}6$ S/M. Clearly, to obtain a bifunctional concentrator, one should make ${\kappa }_{\text{B}}>{\kappa }_0$ and ${\sigma }_B>{\sigma }_0$. Then, one can obtain: ${\kappa }_{\theta }=0.3$, ${\kappa }_r\approx 0.5{\kappa }_B$, ${\sigma }_{\theta }=0$, ${\sigma }_r=0.5{\sigma }_B$. Obviously it is hard to find material B to satisfy the aforementioned conditions. Fortunately, these two conditions can be simplified in the following way. Simulations were conducted to investigate the concentrator’s performance with a series of ${\kappa }_r$and ${\kappa }_{\theta }$ by setting ${\kappa }_{\theta }/{\kappa }_0=0.3/{\kappa }_0=0.02$, ${\sigma }_{\theta }/{\sigma }_{\text{0}}=0$. The thermal concentrator’s performance is given in Fig. 3(a)-3(i) , where a nearly perfect thermal concentrator can be obtained when $\text{m}={\kappa }_r/{\kappa }_0>6$. Quantitatively, we evaluate the thermal concentrator performance by calculating the standard deviation (STD) [32] of temperatures along the black line x = 31mm with different values of m (see Fig. 4 ). Clearly, a smaller value of STD means that smaller distortion and an efficient concentrator means that STD is close to 0. As one can see in such picture, the concentrator performance is enhanced with increased value of m. We also find that when m>6, the concentrator performance changes little and a nearly perfect concentrator can be obtained. Similarly, the concentrator’s performance for electric field is shown in Fig. 5(a)-5(i) , where a nearly perfect electric concentrator can be obtained when $n={\sigma }_r/{\sigma }_0>\text{6}$. Quantitatively, we evaluate the electric concentrator performance by calculating the standard deviation (STD) of potential along the red line x = 31mm with different values of n (see Fig. 6 ). Similarly, it is found that when n>6, the concentrator performance changes little and a nearly perfect electric concentrator can be obtained. Therefore, to make a bifunctional concentrator, the following relations should be met: ${\kappa }_B>12{\kappa }_0=180$W/mK, ${\sigma }_B>12{\sigma }_0=2.4e7$ S/M. This is an important property, which features two desirable aspects: simplified fabrication and easy availability of many natural materials. Aluminum, silver, copper can all be used in our scheme. We chose aluminum with thermal conductivity ${\kappa }_{\text{B}}=237$ W/mK and electric conductivity ${\sigma }_{\text{B}}=3.57\text{e}7$S/M.

 

Fig. 3 Thermal flux distribution for different values: (a) ${\kappa }_r\text{=}{\kappa }_{\text{0}}$, (b) ${\kappa }_r\text{=2}{\kappa }_{\text{0}}$, (c) ${\kappa }_r\text{=3}{\kappa }_{\text{0}}$,(d) ${\kappa }_r\text{=4}{\kappa }_{\text{0}}$, (e) ${\kappa }_r\text{=5}{\kappa }_{\text{0}}$(f) ${\kappa }_r\text{=6}{\kappa }_{\text{0}}$(g) ${\kappa }_r\text{=7}{\kappa }_{\text{0}}$(h) ${\kappa }_r\text{=8}{\kappa }_{\text{0}}$ (i) ${\kappa }_r\text{=10}{\kappa }_{\text{0}}$. The white lines represent isothermal lines. The black line in (a) represents observed line.

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Fig. 4 The calculated standard deviation (STD) of the isotherms at the observed line (along the black line in Fig. 3(a)) with different values of m.

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Fig. 5 Current density distribution for different ${\sigma }_r$value: (a) ${\sigma }_r\text{=}{\sigma }_{\text{0}}$, (b) ${\sigma }_r\text{=2}{\sigma }_{\text{0}}$, (c) ${\sigma }_r\text{=3}{\sigma }_{\text{0}}$,(d) ${\sigma }_r\text{=4}{\sigma }_{\text{0}}$, (e) ${\sigma }_r\text{=5}{\sigma }_{\text{0}}$(f) ${\sigma }_r\text{=6}{\sigma }_{\text{0}}$(g) ${\sigma }_r\text{=7}{\sigma }_{\text{0}}$(h) ${\sigma }_r\text{=8}{\sigma }_{\text{0}}$(i) ${\sigma }_r\text{=10}{\sigma }_{\text{0}}$. The black lines represent isopotential lines. The red line in (a) represents observed line.

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Fig. 6 The calculated standard deviation of the potential at the observed line (along the red line in Fig. 5(a)) with different values of n.

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3. Simulation and experimental results

To confirm our prediction, simulations based on COMOSOL were carried out to obtain the concentrator’s thermal and electric conductivity properties. First, the simulation results for the homogeneous background material are given in Fig. 7(a)-7(b) . As shown in these pictures, a uniform temperature gradient is formed from high temperature (80°C) to low temperature (0°C). Meanwhile, a uniform electric potential gradient also appears from high potential (1V) to low potential (0V). The results for our bifunctional device are shown in Fig. 7(c)-7(d), where both thermal and electric fields are concentrated into the core region, leading to increased temperature and electric potential gradients, at the same time, the external fields are nearly undistorted, indicating good thermal and electric performance for the concentrator. Note that the slight distortion for thermal field and electric field can be attributed to the discrete wedges of fan-shaped structure. The performance can be enhanced by increasing the number of wedges.

 

Fig. 7 Simulation results. (a) Temperature profile for homogeneous background material. (b) Electric potential distribution for homogeneous background material. (c) Temperature profile for bifunctional concentrator. (d) Electric potential distribution for bifunctional concentrator. The white lines represent isothermal lines or isopotential lines.

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In our experiment, such device was fabricated, shown in Fig. 2(c). In thermal measurement, the two sides of sample touched hot water (80°C) and ice water (0°C), respectively. An infrared heat camera (Fluke Ti300) was used to obtain the temperature profile distribution. After several minutes, the steady temperature profile was generated, shown in Fig. 8 . Clearly, the temperature gradient in the core region is increased and the external field remains nearly undistorted, suggesting good thermal performance for the concentrator. To evaluate the electric concentrator’s performance, according to the previous work [16], the gradual structure is used and the potential distribution is measured along lines $x=31$ mm, $x=-31$ mm and $y=0$ mm (see inserts in Fig. 9(a)-9(c) ). As expected, the potential distribution along the lines $x=31$ mm and $x=-31$ mm are straight (Fig. 9(a)-9(b)), which means no distortion occurs. Meanwhile, the potential gradient in the core region is apparently increased (Fig. 9(c)). The measured results are also given in those pictures, indicating good agreement with the simulation. Note that the slight distortion for thermal field and electric field can be attributed to the discrete wedges of fan-shaped structure. The performance can be enhanced by increasing the number of wedges.

 

Fig. 8 Experimental measured temperature profile for the bifucntional device. The black circles represent the inner and outer radii of concentrator shell.

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Fig. 9 The simulation and experiment results for the different cases at corresponding positions: (a) $x=-31$, (b) $x=31$ mm and (c) $y=0$ mm. The white lines in inserts represent observed lines.

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4. Conclusion

In conclusion, we have proposed and given an experimental demonstration of bifunctional concentrator which can simultaneously concentrate both electric and thermal fields into a given region while keeping the external fields undistorted. In the process, only natural materials are used, greatly simplifying the fabrication. The simulation and experimental results show good agreement, indicating the soundness and feasibility of our scheme. This is a general method for realizing bifunctional concentrator, which can also be extended to other multi-physics systems.